Article
wavefunctions, similar to ref.^48 , obtaining αs(v = 0, L = 1) = 3.5054,
αt(v = 0, L = 1) = −0.955, αs(v = 0, L = 0) = 3.4961 and αt(v = 0, L = 0) = 0, in
atomic units. The vector polarizabilities are negligible. The computed
light shift is of the order of 0.01 Hz. We therefore set the correction due
to the 313-nm wave intensity to zero.
Line pulling. We have no observational evidence that Zeeman compo-
nents, or micromotion-induced sidebands of other hyperfine compo-
nents, could affect the measured transitions. The small linewidths of
the measured transitions are important in this respect. We did not
observe any change of f (^16) +, f (^16) − and f 160 at the 10-Hz level upon a 500-Hz
change of the trap frequency.
d.c. offsets. For every measurement reported in the manuscript, the
HD+ ions are located along the symmetry axis of the Be+ ion cluster. An
offset of 10 V was applied to an electrode to displace the beryllium
crystal by about 100 μm from the trap axis along the radial direction.
We observed that this offset potential does not have an effect on the
position of the HD+ ions, as also found in molecular dynamics simula-
tions^16. We measured the frequency shift of f (^19) − caused by this offset
potential to be 1(10) Hz. Possible day-to-day variations of the trap com-
pensation voltage are a small fraction of the applied offset. Therefore,
the size and uncertainty resulting from these variations are negligible.
Light shift due to the two REMPD lasers. The shift due to the 1.4-μm
laser and 266-nm laser waves present during spectroscopy has been
determined by performing spectroscopy in a different mode, alternat-
ing terahertz irradiation and REMPD laser irradiation. The shift has
been measured for all lines and all Zeeman components discussed here.
The shifts are smaller than or equal to 0.039(17) kHz in absolute value.
The measured shifts and their uncertainties are used as corrections.
Other shifts. According to theoretical calculations, the black-body
radiation shift^48 and the molecular electric quadrupole shift^51 can be
neglected at the present level of accuracy.
Data analysis
Extrapolation of the measured frequencies to zero magnetic field
and zero trap amplitude is done by a standard least-squares method.
Standard formulae for the propagation of uncertainties are applied.
Spin coefficients, their uncertainties, and sensitivity of the
transition frequencies to the spin coefficients
To allow for an accurate comparison between experiment and ab initio
theory, we performed a substantially more accurate computation of
the spin-structure coefficients of HD+ compared with our earlier work^35.
We extended the approach developed in ref.^38 and the relevant matrix
elements were calculated to ten significant digits. Values of the two
spin-structure coefficients for the lower level, E 4 and E 5 , and the
nine coefficients for the upper level, E′, ..., 19 E′ are reported in the
Extended Data Table 1. Using these coefficients in the diagonalization
of the spin-structure Hamiltonian of ref.^35 , we obtain the spin frequen-
cies fspin,i (Extended Data Table 1).
The largest spin-structure coefficients, E 4 , E′ 4 , E 5 and E′ 5 , have theo-
retical fractional uncertainties of approximately ε 4 ≈ ε 5 ≈ 1 × 10−6 = εF.
This estimate is confirmed by comparison of the theoretical predictions
of the molecular ion H 2 +, calculated with the same theoretical approach,
with the experimental results of refs.^17 ,^39. For a given vibrational level,
the rotational dependence of the neglected terms in E 4 and E 5 is nearly
zero, because these are contact terms determined by the electronic
wave function, which depends very weakly on N. This allows us to
assume that the neglected terms in (E 4 , E′ 4 ) and in (E 5 , E′ 5 ) are essentially
equal, respectively.
Under this assumption, the theory uncertainty of a spin frequency
due to these coefficients k = 4, 5 is set to uk = |γγ′′ik,,EEk−|ikkεF, where
γi,k = −∂fspin,i/∂Ek is the derivative of the spin energy of the lower quan-
tum state involved in the transition i with respect to the spin coefficient
Ek, and γfik′=,s∂/pin,i∂′Ek is defined analogously for the upper state. The
values of the derivatives are presented in Extended Data Table 1.
The spin Hamiltonian coefficients E 4 ≈ E′ 4 and E 5 ≈ E′ 5 are similar for
the two rotational states, and because the transitions studied here are
those between similar spin states, for which G 11 =′G, G 22 =′G, the spin
frequencies are small, ||fspin,i≪EEE 45 ,,′, 45 E′ and the sensitivities are
similar, γγ′≈ik,,ik. Therefore, we benefit from important reduction of
the theory uncertainties u 4 and u 5 contributed by these four coeffi-
cients. Even in the least favourable case, line 14, the uncertainty con-
tribution is less than or equal to u 4 + u 5 ≈ 14 Hz (1 × 10−11), that is,
negligible compared with the following contributions.
A second set of coefficients, E′ 1 , E′ 6 and E′ 7 , are one to three orders
smaller in magnitude, and have estimated fractional uncertainties of
ε 1 ≈ ε 6 ≈ ε 7 ≈ α^2 = ε 0 ≈ 5 × 10−5. Their absolute uncertainties, 1.5 kHz to
0.06 kHz, are at a relevant level. They enter the spin-structure frequency
uncertainty with contributions uεkk=′E 0.
The fractional uncertainties of the coefficients E′ 2 , E′ 3 , E′ 8 and E′ 9 are
similar to ε 0 , but are not relevant at the present experimental accuracy
level because the coefficients themselves are much smaller than the
others.
As the details of the theory errors are unknown, the total uncer-
tainty of the spin frequencies is set conservatively as the sum over all
uk (instead of the root sum of squares).
The sensitivities γ are obtained by first computing the eigenvalues
Espin,i and E′spin,i of the Hamiltonian analytically and then computing
analytically their derivatives with respect to the individual coefficients
Ek and E′k. These derivatives are then evaluated for the set of current
theory values for Ek and E′k.
Fit of the spin Hamiltonian coefficients
From the six measured transitions, we can derive information about
the spin Hamiltonian coefficients and about the true spin-averaged
frequency. Under the previous assumption of equal theory errors for
(E 4 , E′ 4 ) and for (E 5 , E′ 5 ), there are six remaining important quantities
(E′ 1 , E′ 4 , E′ 5 , E′ 6 , E′ 7 and fspin-avg), and they can be solved for using a set of
equations in which the experimental frequencies are equal to the
corresponding theoretical frequencies, allowing for small deviations
from the nominal values. We find E 1 ′(fit)−E′ 1 (theor)=0.32(20)kHz, where
the uncertainty is smaller than the theory uncertainty, εF1E′≈1.6kHz.
Furthermore, E′ 6 (fit)−E′ 6 (theor)=0.5(9)kHz, E 7 ′−(fit) E 7 ′=(theor) −0.3(4)kHz
and ffsp(fit)inavg−=sp(theinorav)g −0.05(22)kHz. The shown uncertainties
result from the experimental errors and the theory error offsp(theinorav)g;
the theory errors of E′ 2 , E′ 3 , E′ 8 and E′ 9 make negligible contributions.
The deviations of E′ 4 and E′ 5 from the nominal values cannot be deter-
mined precisely (an aspect that is intrinsic to the favoured transitions),
but are consistent with zero.
Composite frequencies
The coefficients of the composite frequency given in the main text are:
bbb
bbb
=0.0 863720 ..., =0.145 6348 ..., =0.2516 111 ...,
=0.2442 79 2...,=0. 132807 4...,=0. 139295 5...
12 14 16
19 20 21
We consider alternative composite frequencies. One alternative
ansatz for finding a composite frequency is to impose the ‘insensitivity
conditions’ 0 =∂fb(tcheor)/∂Eki=∑ iγik,
α α
, 0 =∂f(tcheor)/∂Ek′β for a suitable
subset {kα, kβ} of spin Hamiltonian coefficients. As discussed above, if
we assume correlated errors for the pair (E 4 , E′ 4 ) and (E 5 , E′ 5 ), then the
largest theory uncertainties arise from E′ 1 , E′ 6 and E′ 7. Four experimen-
tally measured transitions are sufficient to satisfy the three insensitiv-
ity conditions for these three coefficients. The normalization condition